Morita's projectivity conjecture for integral models of Shimura varieties

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Let (G,X)(G,\mathcal{X}) be the Shimura datum under consideration, let Gad⁡G^{\operatorname{ad}} be its adjoint group, let N∈NN\in\mathbb{N} with N≥3N\geq 3, and let N(N)\mathcal{N}(N) be the integral model over O(G,X)[1N]O(G,\mathcal{X})[{1\over N}]. Morita's conjecture. If the Q\mathbb{Q}-rank of Gad⁡G^{\operatorname{ad}} is 00, then N(N)\mathcal{N}(N) is projective over O(G,X)[1N]O(G,\mathcal{X})[{1\over N}]. This conjecture predicts projectivity of the integral model in the anisotropic adjoint case, paralleling projectivity of the complex Shimura variety. Its resolution status is not specified in the supplied text.

References

Primary source

Adrian Vasiu, “Geometry of Shimura varieties of Hodge type over finite fields”, arXiv:0712.1840 (2007).

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